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Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.
3

%I #30 Jul 18 2022 16:38:19

%S 0,1,9,39,116,275,561,1029,1744,2781,4225,6171,8724,11999,16121,21225,

%T 27456,34969,43929,54511,66900,81291,97889,116909,138576,163125,

%U 190801,221859,256564,295191,338025,385361,437504,494769,557481,625975,700596

%N Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.

%C a(n) is also the number of semistandard Young tableaux over all partitions of 4 with maximal element <= n. - _Alois P. Heinz_, Mar 22 2012

%C Starting from 1 the partial sums give A244864. - _J. M. Bergot_, Sep 17 2016

%H Alois P. Heinz, <a href="/A139594/b139594.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = coefficient of x^4 in 1/((1-x)^n * (1-x^2)^binomial(n,2)).

%F a(n) = (n^2*(7+5*n^2))/12. G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^5. [_Colin Barker_, Mar 18 2012]

%e From _Michael B. Porter_, Sep 18 2016: (Start)

%e The nine 2 X 2 matrices summing to 4 are:

%e 4 0 3 0 2 0 1 0 0 0 2 1 1 1 0 1 0 2

%e 0 0 0 1 0 2 0 3 0 4 1 0 1 1 1 2 2 0

%e (End)

%p dd := proc(n,m) coeftayl(1/((1-X)^m*(1-X^2)^binomial(m,2)),X=0,n); seq(dd(4,m),m=0..N);

%t gf[k_] := 1/((1-x)^k (1-x^2)^(k(k-1)/2));

%t T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];

%t a[k_] := T[4, k];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 07 2020 *)

%Y For 3 in place of 4 this gives A005900.

%Y Row n=4 of A210391. - _Alois P. Heinz_, Mar 22 2012

%Y Partial sums of A063489.

%K easy,nonn

%O 0,3

%A _Marc A. A. van Leeuwen_, Jun 12 2008